This text was used in my History of Mathematics class in Spring 2005, which aimed to cover Part 1 (Chapters 1-8).
Even some non-human animals have a basic concept of number, and we have archaeological evidence that humans developed counting as far back as 50,000 years ago. The simplest system of numerals, the tally system, developed from this, on the concept of a one-to-one correspondence between marks, notches, knots, or other features and the objects to be counted.
Early number words differed according to the type of object counted, such as two sheep or two men. For example, "team of horses, span of mules, yoke of oxen".
The given example (rendered partly above) seems to refer to a generic grouping, rather than a specific number of a specific type of thing. Were there four distinct words for two or seven men or sheep?
Various number bases (or radixes or scales) have been used throughout history, including 2, 3, 4, 5 (quinary scale), 12 (duodecimal scale), 20 (vigesimal scale), and 60 (sexagesimal scale), as well as, of course, 10.
Finger numbers (whence the word digit) were once widely used (and are still rarely used), but eventually these developed into written numerals.
The simplest type of numeral system is a simple grouping system, in which the values of symbols are combined purely additively. The Egyptians used such a system, and the Babylonians did as well, extended with a subtractive component.
Other cultures, such as the Chinese and Japanese, used a multiplicative grouping system. For example, 5625 is written 五千六百二十五 in Japanese.
The Ionic, or alphabetic, Greek numeral system is a ciphered numeral system, in which the various letters of the Greek alphabet stand for numbers up to 900 (so three letters might be used to write a three-digit number, but the characters for 20 and 200 differ: they are kappa and sigma).
Our own Arabic numeral system is a positional numeral system. The Babylonians developed a system of this sort between 3000 and 2000 BC. The earliest example of the Hindu-Arabic numeral system is found in India, and dates to about 250 BC, but do not feature a symbol for zero or a positional system. These appeared by AD 800.
The Mayans used a peculiar mixed-base positional system.
Even with our modern numeral system, modern-style computation would not be practical without paper or similar writing material, which was too precious in the ancient world for such methods to develop. The Egyptians created papyrus from a water reed called papu. Parchment, made from animal skins, was also used, but this was so expensive that it was washed and reused. Manuscripts written on such parchment are called palimpsests (from palin, "again"; psao, "rub smooth"). The Romans wrote on waxed boards about 2000 years ago, and sand trays were also used for temporary counting and figuring.
To compensate for the lack of paper, the abacus (Greek abax, "sand tray") was invented, which allows rapid computation and is, importantly, reusable.
By AD 1500, modern-style computation had won out over the use of the abacus.
The use of the abacus has continued in Japan until the present day, although its use is decreasing, with the ubiquity of computers.
The algorithms we use for computation of sums, products, etc. are actually independent of the base of the numbers. By using a table or simply working out the intermediate results, the same algorithm may be employed to produce results in any base. Of course, the speed with which we accomplish figuring in base 10 is due to our memorization of these tables, so working in other bases will be much slower unless similar fluency is obtained.
Most of the problem studies are just calculations, or are otherwise uninteresting. I'll give a couple of samples.
Explain why certain number words might be used. For example:
(c) The South American Kamayura tribe uses the word peak-finger as their word for 3, and "3 days" comes out as "peak-finger days."
The obvious explanation is that the third finger is the longest. Perhaps the other number words are also based on the fingers.
Make use of the various numeral systems described in the chapter. For example:
(i) Record 4 times XCIV in Roman numerals.
4*94 = 376 = CCCLXXVI
On p. 38 are references arguing for religious ritual as the origin of mathemaics.
This chapter is limited to discussion of mathematics in Babylonia and Egypt, because not enough information is preserved about ancient Chinese and Indian mathematics, or presumably mathematics from any other regions.
Do we, today, in fact know little of these things? Have we learned more, or is there perhaps an oversight in the book?
Babylonian mathematics had an algebraic character, the problems stated in prose.
Egyptians represented all fractions, except 2/3, as the sum of unit fractions, which have unit numerators.
Egyptians solved linear equations using the rule of false position:
Egyptian mathematics made some use of symbols, including ideograms for plus, minus, equals, and unknown.
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